Decoherence of multicomponent and multimode generalizations of even/odd coherent states in thermal and phase reservoirs

Abstract

Within the framework of the standard master equation for a quantum damped harmonic oscillator interacting with a phase-insensitive (thermal) reservoir, we study the decoherence of superpositions of displaced quantum states of the form \(\sum\nolimits_{k = 1}^N {c_k \hat D(\alpha _k )\left. {|g} \right\rangle } \), where |g〉 is an arbitrary “fiducial” state and \(\hat D(\alpha )\) is the usual displacement operator. We compare two simple measures of degree of decoherence — the quantum purity and the height of the central interference peak of the Wigner function. We show that for N > 2 components of the superposition, the decoherence process cannot be characterized by a single decoherence time if |α| ≫ 1. Therefore, we distinguish the “initial decoherence time” (IDT) and “final decoherence time” (FDT) and study their dependence on parameters α k and N. Explicit exact expressions are obtained in the special case of |g〉 = |m〉, i.e., for (symmetric) superpositions of displaced number states. We show that the superposition with a big number of components N and rich “internal structure” (m ∼ |α|2) can be more robust against decoherence than simple superpositions of two coherent states, even if the initial decoherence times coincide. Comparing the decoherence of n-mode superpositions of coherent states, we show that the FDT of initially factorized states can be significantly bigger than that of initially maximum entangled states with the same initial energy, especially if n ≫ 1. We compare the decoherence rates of even/odd superpositions of displaced number states |m, α〉± and photon-added coherent states |α, m〉± and show that their dependence on m is completely different. Finally, we find analytical expressions for the Wigner function and the time-dependent purity of even/odd coherent states in the case of phase damping. In this case, the IDT has the same dependence on the distance between the two components of the superposition as in the case of amplitude damping. However, the long-time behavior of the Wigner function is quite different, because the asymptotical stationary state is not a thermal one, but a strongly nonclassical (although highly mixed) state.